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Finite-dimensional alternative (Z_2xZ_2-graded) Lie superalgebras are discussed parallel with the finite-dimensional standard (Z_2-graded) Lie superalgebras. We also consider applications for some of these superalgebras as the relativistic supersymmetries (SUSY). We show that every standard relativistic SUSY (super-anti- de-Sitter, super-Poincar`e, super-conformal, etc.) has an alternative variant. We also show that alternative (Z_2xZ_2-graded) relativistic SUSY has different algebraic structure of the superspace and the representation theory with respect to the standard (Z_2-graded) relativistic SUSY.